2. Robert: Motivation Principles of Optics Applications Optimization Andy: Materials Loss vs. amplification Theoretical problems Overview 2 + 2 = 4WM

3. Motivation 1.Internet relies on fiber optics. 2. Amplification needed. Current technology inadequate:  Limited amplification bandwidth  Limited internet speed

4. Linear Optics Low intensity light in transparent media. Refraction Dispersion Light slows down in transparent media. Refractive index is function of frequency.

5. Propagation constant Beta (propagation constant) very useful. Expressible as power series. Coefficient critical to optimizing FWM.

6. Limitations 1.Photons do not interact. 2.No new frequencies are created. 3.Too simple for our purposes. But nonlinear optics provides us with great possibilities…

7. Nonlinear Optics Kerr effect: refractive index depends on intensity of light. Nonlinearity causes complex behavior. Nonlinear Schrödinger Equation Photons can mix and change their frequencies! Nonlinear Term

8. Four-wave Mixing Signal Pump Lasers Photons added to signal Photons added to idler Idler (created to conserve energy) (Amplified through FWM) 2 + 2 = 4WM Frequency (ω)/100THz Log(Intensity) Pump photons mix to form signal and idler photons.

9. Elastic Collision Analogy Energy Conservation: Momentum Conservation: Pump energies Energies of signal and idler Pump momenta Momenta of signal and idler

10. Applications What can we use it for? Amplification and Frequency conversion. Solves world hunger (for internet speed) Optimization: How do we turn ideas into high performance technology?  mathematical analysis and approximation.

11. Amplification Optimization Amplification depends on only one number. Must be close to – γ P for maximum gain. Complexity of β solved by quartic approximation.

12. Conditions for Maximum Flat Gain 0 1.Average pump frequency at zero dispersion point ω 0. Where: 4 2.β 4 must be positive. 3.And lastly, regarding the pumps:

13. Before and After Optimization Signal Frequency Offset Gain Inferior bandwidth Optimized bandwidth

14. Frequency Conversion Optimization Idler photons used as new signal: Useful since different frequencies needed in fiber. Problem: pumps:  same average frequency as “a” and “b.”  Stuck with bandwidth we’re given…

15. Dispersion Engineering Optical fibers:  Total internal reflection Light strays into cladding. Samples 2 refractive indices. 2, 3, 4 etc. We can engineer β 2, β 3, β 4 etc. 2n21n12n21n1

16. Frequency Conversion Optimization Idler photons used as new signal: Useful since different frequencies needed. Problem: pumps:  same average frequency as “a” and “b.”  Stuck with bandwidth we’re given… 4 Solution: dispersion engineering:  minimize β 4 3 4  Make β 3 and β 4 into “magic ratio.” Creates greater bandwidth.

17. Summary

18. Optical Fiber  Nonlinear effect ∝ γPL  Silica  Low loss  Low nonlinearity γ  High P and L needed for FWM

19. Silica V.S. Chalcogenide SilicaChalcogenide Made of SiO 2 S, Se, Te +others γLowHigh Used inOptical FiberOptical Chip LossLowHigh

20. Nonlinear Schrodinger Equation (NLS)  Linear loss coefficient  Numerically solve NLS with loss (Split step Fourier method)  How loss affects gains

21. 1 pump case Signal Gain Idler Gain γ+γ+ γ-γ- INPUT OUTPUT

22. 1 pump case INPUT OUTPUT

23. Gain curve – 1 pump

24. α = (dB/m) 0 20 40 60 80 100 Chalcogenide

25. Peak Gain– 1 pump loss ∝ e -αL (dB/m)

26. 2 pump case INPUT OUTPUT Signal GainIdler Gain 2 pump case

27. Gain curve - 2 pump case Signal Gain Idler Gain

28. Asymmetry Problem

29. Conclusion  FWM : nonlinear optical effect  Parametric amplifications  Conditions for greater bandwidth  How loss affects gain curves— unexpected!!

30. Future Work  Asymmetry Problem  Coping with loss

31. References C. J. McKinstrie, S. Radic and A. R. Chraplyvy. Parametric Amplifiers Driven by Two Pump Waves. IEEE J. Quantum Electron., vol.QE-8, pp. 538–547, 2002. G. P. Agrawal (2001). Nonlinear Fiber Optics. Orlando: Academic Press. M. R. Lamont, T. T. Kuhlmey and C. M. de Sterke. Multi-order dispersion engineering for optimal four-wave mixing. Optics Express, vol.16, pp. 7551–7563, 2008.

32. Thanks for listening